Approximately Efficient Two-Sided Combinatorial Auctions · One-sided markets have been studied in...
Transcript of Approximately Efficient Two-Sided Combinatorial Auctions · One-sided markets have been studied in...
Approximately E�icient Two-Sided Combinatorial Auctions
RICCARDO COLINI-BALDESCHI, Dipartimento di Economia e Finanza, LUISS
PAUL W. GOLDBERG, University of Oxford
BART DE KEIJZER, Centrum Wiskunde & Informatica (CWI)
STEFANO LEONARDI, Sapienza, University of Rome
TIM ROUGHGARDEN, Stanford University
STEFANO TURCHETTA, KPMG Italy and University of Oxford
We develop and extend a line of recent work on the design of mechanisms for two-sided markets. �e markets
we consider consist of buyers and sellers of a number of items, and the aim of a mechanism is to improve
the social welfare by arranging purchases and sales of the items. A mechanism is given prior distributions
on the agents’ valuations of the items, but not the actual valuations; thus the aim is to maximise the expected
social welfare over these distributions. As in previous work, we are interested in the worst-case ratio between
the social welfare achieved by a truthful mechanism, and the best social welfare possible.
Our main result is an incentive compatible and budget balanced constant-factor approximation mechanism
in a se�ing where buyers have XOS valuations and sellers’ valuations are additive. �is is the �rst such
approximation mechanism for a two-sided market se�ing where the agents have combinatorial valuation
functions. To achieve this result, we introduce a more general kind of demand query that seems to be needed
in this situation. In the simpler case that sellers have unit supply (each having just one item to sell), we give
a new mechanism whose welfare guarantee improves on a recent one in the literature. We also introduce
a more demanding version of the strong budget balance (SBB) criterion, aimed at ruling out certain “unnatural”
transactions satis�ed by SBB. We show that the stronger version is satis�ed by our mechanisms.
Additional Key Words and Phrases: Auctions; mechanism design; market intermediation; two-sided markets;
pricing; approximation algorithms
1 INTRODUCTIONOne-sided markets have been studied in economics for several decades and more recently in com-
puter science. Mechanism design in one-sided markets aims to �nd an e�cient (high-welfare)
allocation of a set of items to a set of agents, while ensuring that truthfully reporting the input data
is the best strategy for the agents. �e cornerstone method in mechanism design is the Vickrey-
Clarke-Groves (VCG) mechanism [4, 13, 23] that optimises the social welfare while providing the
right incentives for truth-telling: VCG mechanisms are dominant strategy incentive compatible(DSIC), and in many mechanism design se�ings VCG is also individually rational (IR). �e IR
requirement demands that participating in the mechanism is not harmful to any agent. �e DSIC
requrement demands that truthfully reporting one’s preferences to the mechanism is a dominant
strategy for each agent, independently of what the other agents report.
Recently, increased a�ention has turned to the problems that arise in two-sided markets, in which
the set of agents is partitioned into buyers and sellers. In contrast with the one-sided se�ing (where
one could say that the mechanism itself initially holds the items), in the two-sided se�ing the items
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h�p://dx.doi.org/10.1145/3033274.3085128
DOI: http://dx.doi.org/10.1145/3033274.3085128
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are initially held by the sellers, who have valuations over the items they hold, and who are assumed
to act rationally and strategically. �e mechanism’s task is now to decide which buyers and sellers
should trade, and at which prices. �e growing interest in two-sided markets can be a�ributed to
various important applications. Relevant examples are selling display-ads on ad exchange platforms,
the US FCC spectrum license reallocation, and stock exchanges. Two-sided markets are usually
studied in a Bayesian se�ing: there is public knowledge of probability distributions, one for each
buyer and one for each seller, from which the valuations of the buyers and sellers are drawn.
In two-sided markets, a further important requirement is strong budget balance (SBB), whichstates that monetary transfers happen only among the agents in the market, i.e., the buyers and
sellers are allowed to trade without leaving to the mechanism any share of the payments, and
without the mechanism adding money to the market. A weaker version of SBB o�en considered
in the literature is weak budget balance (WBB), which only requires the mechanism not to inject
money into the market. However, it is known from the work of Myerson and Sa�erthwaite [17]
that it is generally impossible for an IR, BIC, and WBB mechanism to maximise social welfare in
such a market, even in the bilateral trade se�ing, i.e., when there is just one seller and one buyer.1
�e practical contexts noted above need the application of two-sided market mechanisms that
can work in a combinatorial se�ing, i.e., where there are multiple distinct items in the market
and agents having possibly complex valuations over the subsets of items that they may receive.
However, we are not aware of any such mechanism that approximates the social welfare while
meeting the IR, DSIC and SBB requirements. �e purpose of this paper is to provide mechanisms
that satisfy these requirements and achieve an O (1)-approximation to the social welfare for a
broad class of agents’ valuation functions. We do, in fact, design mechanisms that work under the
assumption of the valuations being fractionally subadditive (XOS), a generalisation of submodular
functions that are contained in the class of subadditive functions.
Our results extend and improve on previous work which targeted an important special case
of a two-sided market: each seller holds a single item, items are identical, and each agent is only
interested in holding a single item. In this se�ing, the valuations of the agents are thus given
by a single number, representing the agent’s utility for holding an item. A mechanism for this
se�ing is known in the literature as a double auction. �e goal of several works on double auctions
[15, 20, 21] has been that of trading o� the achievable social welfare with the strength of the
incentive compatibility and budget balance constraints. In our present work, we investigate this
question for the much more general class of combinatorial two-sided markets.
1.1 The ModelAs stated above, the set of agents is partitioned into a set of sellers, each of which is initially endowedwith a set of heterogeneous items, and a set of buyers, having no items initially. Buyers have money
that can be used to pay for items. Every agent has its own, private valuation function, which maps
subsets of the items to numbers, and agents are assumed to optimise their (quasi-linear) utility,
which is given by the valuation of the set of items that the mechanism allocates to an agent, minus
the payment that the mechanism collects from the agent. A seller will typically receive money
(instead of pay money), which we treat as a negative payment.
For each agent we are given a (publicly known) probability distribution over a set of valuation
functions, from which we assume her valuation function is drawn. �e mechanism and the other
1�e VCG mechanism can also be applied to two-sided markets; however, in this se�ing, VCG is either not IR or it does
not satisfy WBB.
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agents have no knowledge of the actual valuation function of the i-th agent, but only of her prob-
ability distribution. �e general aim of the mechanism is to reallocate the items so as to maximise
the expected social welfare (the sum of the agents’ valuations of the resulting allocation).
Let OPT be the expected social welfare of an optimal allocation of the items. Note that this is
a well-de�ned quantity, even though computing an optimal allocation may be computationally
hard, and even though there might not exist an appropriate mechanism (satisfying IR, SBB, and
DSIC), that is guaranteed to always output an optimal allocation.
We are interested in mechanisms that satisfy IR, SBB, and DSIC (or failing that, the weaker
notion of Bayesian incentive compatibility (BIC)), and that reallocate the items in such a way that
the expected social welfare is within some constant fraction of OPT, where expectation is taken
over the given probability distributions of the agents’ valuations, and over the randomness of
the allocation that the mechanism outputs. In contrast with one-sided combinatorial auction
design (where the main challenge is polynomial-time implementability), for the two-sided case our
primary goal is to design (and thus show the existence of) IR, SBB, and DSIC/BIC mechanisms that
O (1)-approximate OPT. Such mechanisms circumvent the aformentioned impossibility result of
Myerson and Sa�erthwaite [17] by weakening the requirement of optimal social welfare to that of
approximately optimal social welfare (while nonetheless strengthening theWBB contraint into SBB).
1.2 Our Results and their Significance�e present paper starts o� by showing that there is a straightforward technical trick that one may
apply to turn any WBB mechanism into an SBB one, with a small loss in the approximation factor.
Technically, one could e.g. apply it to the WBB mechanism of Blumrosen and Dobzinski [1] for
combinatorial exchange markets; however, the trick is unsatisfactory in practice as it essentially
consists of giving the le�over money to a random agent. �is demonstrates a weakness in the
current de�nition of SBB, which motivates the introduction of a strengthened version, that we call
direct-trade strong budget balance (DSBB).Our goal is the design of individually rational, incentive compatible, and direct-trade strongly
budget balanced mechanisms for combinatorial two-sided markets, that achieve a constant approx-
imation to the optimal social welfare. We present two mechanisms adhering to these constraints
for general families of combinatorial two-sided markets, as summarized in the table below.
MechanismBuyers’
valuations
Sellers’
valuations
Approximation
ratioIR IC BB
M1-supply XOS unit-supply 6 ex-post IR DSIC DSBB
Madd XOS additive 6 interim IR BIC DSBB
Madd additive additive 6 ex-post IR DSIC DSBB
Table 1. Summary of our results.
OurM1-supply mechanism handles the se�ing where all sellers have a single item for sale, and
buyers have fractionally subadditive (XOS) valuation functions over the set of items in the market.
OurMadd mechanism can handle the more general case where sellers have multiple items for sale
and have additive valuation functions over the items they possess, thoughMadd satis�es weaker
IC and IR notions than M1-supply. More precisely, Madd is DSIC and IR on the sellers’ side and
BIC and interim-IR on the buyers’ side. However, for the special case where buyers have additive
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valuation functions,Madd does satisfy the stronger IC and IR notions for both buyers and sellers.
In all three cases, DSBB is satis�ed (a strengthened variant of SBB), and our mechanisms achieve
an O (1)-approximation to the optimal social welfare.
To our knowledge, these are the �rst mechanisms for combinatorial two-sided markets that
simultaneously are IC, (D)SBB, IR, and approximate the optimal social welfare to within a constant
factor. Notice that with non-unit-supply sellers, a constant approximation was not previously
known even in the context of WBB or standard SBB.2Furthermore, we note that our mechanisms
not only work for a more general se�ing than that of [5], but also improve the approximation ratio
for double auctions from 16 to 6.
In the case ofMadd, buyers are required to answer a generalised type of demand query, in which
the mechanism gives prices for the items, and asks a buyer which bundle she would like if, foreach item in that bundle, she were to receive it with probability 1/2. Our usage of these queriescould be criticised for imposing an excessive cognitive burden on the agents. Although we are not
concerned here with that issue (we model agents as computationally unbounded as well as rational),
our apparent need for such queries highlights the general question of how agents’ computational
limitations a�ect what outcomes can be achieved.3
1.3 Overview of the Techniques�e main challenge in two-sided market design is to �nd prices that stimulate truthful behavior
and are suitable for both buyers and sellers, which have contrasting interests. In fact, even in the
simplest imaginable se�ing – the bilateral trade – it is impossible to design a socially e�cient
mechanism satisfying IR, BIC and WBB [17].
A �rst feature all our mechanisms share to guarantee DSBB is being a generalised version
of two-sided sequential posted price mechanisms (SPMs) [5] for double auctions to combinatorial
two-sided markets. �ese mechanisms assign �xed, pre-computed prices to each item so that these
prices are the only ones for which the items can be traded. �is yields a sequence of bilateral trades
in which the amount paid by the buyer equals the amount received by the seller.
While one-sided SPMs provide IR and IC for free, two-sided SPMs require additional conditions
to be met. In combinatorial two-sided markets, if prices are �xed for every single item, it cannot
be guaranteed that a bundle of items chosen by a buyer will surely be allocated to her, in case at
that point the corresponding seller has not been queried yet about her willingness to sell the item.
Symmetrically, when a seller would communicate to an SPM mechanism which bundle of items she
is willing to sell given the proposed item prices, then the mechanism cannot guarantee to the seller
that this bundle will surely be traded in case it has not yet queried the buyers which item sets they
demand. �e situation is further complicated by the fact that there may exist strong interdependen-
cies among items within an agent’s valuation function, which implies that the choice of bundle that
a buyer requests (or that a seller makes available) depends strongly on the set of items that the sellers
are prepared to sell (or that the buyers request). �erefore, a mechanism designer needs to be careful
in proposing prices that are suitable for both sides of the market, and needs to be particularly careful
in selecting the side of the market to process �rst. �e choice that the mechanism made here can de-
pend crucially on the types of valuation functions of the agents. Indeed, onemain di�erence between
our two mechanisms is the order in which we process each side of the market. Anyway all the mech-
anisms proposed in this paper are oblivious to the order in which sellers and buyers are presented.
2�e mechanism proposed in [1] achieves a constant approximation to the optimal social welfare if the size of the initial
endowment of each agent is bounded by a constant; otherwise the approximation factor is of logarithmic order.
3�is question also applies to standard demand queries [10], which may be computationally hard to answer or may involve
a high communication complexity, depending on the computational model used and on the way in which the valuation
functions are represented.
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To additionally achieve a mechanism that results in a high social welfare, we exclude some
items from trade and introduce randomness into the mechanism. �e main idea is to suppose
that all the items are available to the set of buyers as in a one-sided auction, and to compute the
expected marginal contribution of an item to the social welfare [11] under this assumption. �en,
the mechanism compares this contribution to the seller’s value for the item: if the seller’s value is
much higher, then we exclude the item from trade and leave it with the seller. �us, the mechanism
only trades items that are of relatively high expected value to the buyers’ side of the market.
To estimate the expected marginal contribution of an item to the social welfare,M1-supply and
Madd make use of an algorithm A that, given a buyers’ valuation pro�le and a set of items, allocates
the items to the buyers, without considering the sellers and their valuations. If one is not interested
in achieving a low runtime, one can take A to be an exact algorithm that outputs an optimal allo-
cation. Alternatively, by using a technique of Feldman et al. [11], one may take for A a polynomial
time approximation algorithm and combine this with sampling a su�ciently number of valuation
pro�les from the distribution, in order to estimate the expected marginal contribution of an item
to the social welfare accurately in polynomial time. �is yields polynomial-time implementable ap-
proximation mechanisms. In particular, in case A is a polynomial time α-approximation algorithm,
it will run within time POLY (1/ϵ ,n,m), and approximate the optimal social welfare within anO (α )multiplicative factor and an ϵ additive term, where ϵ is a parameter that results from the sampling
procedure. �is technique is described in further detail in [11] and works for distributions with
bounded support.
Randomness is added to make sure every seller independently sells her bundle of items with a
�xed probability of 1/2; which is used to bound the social welfare loss on both sides of the market
by no more than a constant factor.
1.4 Related WorkDue to the impossibility result of Myerson and Sa�erthwaite [17], no two-sided mechanism can
simultaneously achieve optimal social welfare and satisfy the BIC, IR, WBB constraints, even in
the simple bilateral trade se�ing. Follow-up work thus had to focus on designing mechanisms that
trade o� among these properties.
�e following papers of the Economics literature studied the convergence rate to the optimal
social welfare as a function of the number of agents when all sellers’ and buyers’ valuations are
independently respectively drawn from identical regular distributions, while satisfying IR and
WBB. Gresik and Sa�erthwaite [12] showed that duplicating the number of agents by τ results in a
market where the optimal IR, IC, WBB mechanism’s expected social welfare approximation factor
approaches 1 at a rate of O (logτ/τ 2). Rustichini et al. [18] and Sa�erthwaite and Williams [21]
investigated a family of non-IC double auctions, and study the ine�ciency and the extent to which
agents misreport their valuations in these double auctions. We remark that these results only hold
for unit-demand buyers and unit-supply sellers, identical valuation distributions, and the hidden
constants in these asymptotic results depend on the speci�c valuation distributions. In contrast,
our interest is in �nding universal constant approximation guarantees for combinatorial se�ings
and not necessarily identical distributions.
In McAfee [15], an IC, WBB, IR double auction is proposed that extracts at least a (1 − 1/`)fraction of the maximum social welfare, where ` is the number of traders in the optimal solution.
Optimal revenue-maximising Bayesian auctions were characterized in Myerson [16], which
provides an elegant tool applicable to single-parameter, one-sided auctions. Various subsequent
articles dealt with extending these results. Related to our work is the work of Deng et al. [6], which
studied maximising the auctioneer’s revenue in Bayesian double auctions. �e same objective was
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studied by Deshmukh et al. [7] yet in the prior-freemodel. In [22], mechanisms for some special cases
of two-sided markets are presented that work by a combination of random sampling and random
serial dictatorship. �e mechanism is IR, SBB and DSIC and its gain from trade approaches theoptimumwhen the market is su�ciently large. Mechanisms that are IC, IR, and SBB have been given
for bilateral trade in [1]. In addition to this, the authors proposed a WBB mechanism for a general
class of combinatorial exchange markets. We will use this result to construct our initial mechanism.
Sequential posted price mechanisms (SPMs) in one-sided markets have been introduced by Sand-
holm and Gilpin [19] and have gained a�ention due to their simplicity, robustness to collusion, and
their easy implementability in practical applications. One of the �rst theoretical results concerning
SPMs is an asymptotic comparison among three di�erent types of single-parameter mechanisms
[2]. �ey were later studied by Chawla et al. [3] for the objective of revenue maximisation. Ad-
ditionally, Kleinberg and Weinberg [14] and Du�ing and Kleinberg [8] strengthen these results
further. Very relevant to our work is the paper of Feldman et al. [11], showing that sequential
posted price mechanisms can approximate social welfare up to a constant factor of 1/2 for XOSvaluation functions if the published price for an item is equal to the expected additive contribution
of the item to the social welfare.
A line of recent work addressed the problem of approximating social welfare in double auctions
and related problems under the WBB requirement. Du�ing et al. [9] indeed proposed a greedy
strategy that combines the one-sided VCG mechanism, independently applied to buyers and to sell-
ers with the trade-reduction mechanism of McAfee [15]. �ey obtain IR, DSIC, WBB mechanisms
with a good approximation of the social welfare, for knapsack, matching and matroid allocation
constraints. More recently, Colini-Baldeschi et al. [5] presented the �rst double auction that satis�es
IR, DSIC, and SBB, and approximates the optimal (expected) social welfare up to a constant factor.
�ese results hold for any number of buyers and sellers with arbitrary, independent distributions on
valuations. �e mechanisms are also extended to the se�ing where there is an additional matroid
constraint on the set of buyers who can purchase an item.
2 PRELIMINARIESAs a general convention, we use boldface notation for vectors and use [a] to denote the set {1, . . . ,a}.We will use I(X ) to denote the indicator function that maps to 1 if and only if event/fact X holds.
2.1 MarketsA two-sided market comprises a set of two distinct types of agents: the sellers, who initially hold
items for sale, and the buyers, who are interested in buying the sellers’ items. All agents possess
a monotone and normalized valuation function, mapping subsets of items to R≥0.4Formally, we
represent a two-sided market as a tuple (n,m,k,I ,G,F ), where [n] denotes the set of buyers, [m]
denotes the set of sellers, [k] denotes the set of all items for sale, I := (I1, . . . , Im ) is a vector of (mu-
tually disjoint) sets of items called the initial endowment, where Ij is the set of items that is initially
held by seller j ∈ [m]. It holds that
⋃mj=1 Ij = [k]. Vectors G = (G1, . . . ,Gn ) and F = (F1, . . . ,Fm )
are vectors of probability distributions, from which the buyers’ and sellers’ valuation functions
are assumed to be drawn: �e valuation function of buyer i ∈ [n] is drawn from distributionGi ,
and similarly the valuation function of seller j ∈ [m] is drawn from distribution Fj .A (combinatorial) exchange market is a more general version of the above de�ned two-sided
market where an agent can act as both a buyer and a seller. �us, everyone may initially own items
4By a monotone valuation function v we mean that v (S ) ≥ v (T ) for all sets of items T ⊆ S . �at is, ge�ing more items
cannot decrease an agent’s overall valuation. By normalized we mean that v (∅) = 0.
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and may both sell and buy items. As a result, in this se�ing, we override the notation and simply
use n to denote the total number of agents. Formally, an exchange market is thus a tuple (n,k,I ,F ).In two-sided markets, sellers are assumed to only value items in their initial bundle and are
therefore not interested in buying from other sellers, i.e., ∀j ∈ [m] and ∀S ⊆ [k],w j (S ) = w j (S ∩ Ij ).Conversely, in exchange markets, no such restriction on the valuation functions exists.
�roughout the paper, we reserve the usage of the le�er i to denote a single buyer, the le�er
j to denote a single seller, and the le�er ` to denote a single item. Moreover, we use vi to denote
buyer i’s valuation function andw j to denote seller j’s valuation function.
2.2 Mechanism Design Goals�e following discussion is speci�c to two-sided markets (the main focus of this paper), but these
concepts can be extended straightforwardly to combinatorial exchange markets. Given a two-sided
market, our aim is to redistribute the items among the agents so as to maximise the social welfare(the sum of the agents’ valuations). An allocation for a two-sided market (n,m,k,I ,G,F ) is a pairof vectors (X ,Y ) = ((X1, . . . ,Xn ), (Y1, . . . ,Ym )) such that the union of X1, . . . ,Xn ,Y1, . . . ,Ym is [k],and X1, . . . ,Xn ,Y1, . . . ,Ym are mutually disjoint. When discussing a given two-sided market, we
will denote by A the set of all allocations for that market.
Redistribution of the items is done by running a mechanismM. A mechanism interacts with and
receives input from the agents, and outputs an outcome, consisting of an allocation (X ,Y ) and a
payment vector (ρB ,ρS ) ∈ Rn × Rm , where ρBrefers to the buyers’ vector of payments and ρS to
the sellers’ one. An outcome is therefore a tuple (X ,Y ,ρB ,ρS ). Note that when an agent is charged
a negative payment, this should be interpreted as an agent receiving money. �e payment of a
seller is usually negative in a reasonable two-sided market mechanism, and this is also the case
for the mechanisms proposed in the present paper.
Agents are assumed to maximise their utility, which is de�ned as the valuation for the bun-
dle of items that they possess with respect to the allocation vector, minus the payment charged
by the mechanism. In particular, the utility uBi (vi , (X ,Y ,ρB ,ρS )) of a buyer i ∈ [n] with valu-
ation function vi is vi (Xi ) − ρBi , whereas for a seller j ∈ [m] with valuation function w j it is
uSj (w j , (X ,Y ,ρB ,ρS )) = w j (Yj ) − ρSj .5
Furthermore, agents are assumed to be fully rational, so that they will strategically interact with
the mechanism to achieve their goal of maximising utility. Our goal is to design a mechanism
such that there is a dominant strategy or Bayes-Nash equilibrium for the agents under which the
mechanism returns an allocation with a high social welfare. For an allocation (X ,Y ), the socialwelfare SW(X ,Y ) is de�ned as
SW(X ,Y ) =∑i ∈[n]
vi (Xi ) +∑j ∈[m]
w j (Yj ).
We now describe three main economic properties our mechanisms must satisfy. For each of these
constraints we �rst introduce the strictest version and then a more relaxed one. We aim to satisfy
the strictest versions, whenever possible.
• Incentive compatibility (IC)6
5Note that Yj represents the bundle of items that remains in the seller’s possession a�er execution of the mechanism.
6Technically, as can be inferred, the DSIC properties are reserved for direct revelation mechansims, i.e., where the buyersolely interacts with the mechanism reporting her valuation function. It is well-known that mechanisms admi�ing a
dominant strategy can be transformed into DSIC direct revelation mechanisms, and those with a Bayes-Nash equilibrium
can be transformed into BIC direct revelation mechanisms. �is way, the DSIC and BIC de�nitions naturally extend to
non-direct revelation mechanisms.
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– Dominant strategy incentive compatibility (DSIC): It is a dominant strategy for every
agent to report her true valuation sincerely. I.e., for every agent i and for every vector
of valuations of all other players, it is impossible for agent i to increase her expected
utility by misreporting her valuation.
– Bayesian incentive compatibility (BIC): It is a Bayes-Nash equilibrium (BNE) for the
agents to truthfully report their valuations to the mechanism. I.e., each agent i max-
imizes her expected utility by truthfully reporting her valuation if all other players
also truthfully report their valuations.
• Individual rationality (IR)– Ex-post individual rationality (ex-post IR): It is not harmful for any agent to participate
in the mechanism, i.e., there is guaranteed to be a strategy for an agent that yields the
agent a utility that is not less than her initial utility. (�e initial utility of a seller with
bundle Ij isw j (Ij ), and the initial utility of a buyer is vi (∅) = 0.)
– Interim individual rationality (interim IR): �ere is a strategy for each agent that yields
her an expected utility that is not less than her initial utility (where expectation is over
the random outcome of the mechanism, resulting from internal randomness of the
mechanism and randomized strategies adopted by the agents).
• Budget balance (BB)– Strong Budget Balance (SBB):�e sum of all agents’ payments output by the mechanism
is equal to zero. Conceptually, this means that no money ends up at an external party,
and no external party needs to subsidise the mechanism.
– Weak Budget Balance (WBB): �e sum of all payments is at least zero. In two sided-
markets, this generally means that the buyers’ payments are at least as large as the
payments received by the sellers. No external party needs to subsidise the mechanism.
For valuation pro�les (v,w ), OPT(v,w ) := max{SW(X ,Y ) : (X ,Y ) ∈ A} denotes the optimalsocial welfare. �e expected optimal social welfare is the value OPT = Ev,w [OPT(v,w )]. We say
that a mechanism M α-approximates the optimal social welfare for some α > 1 if and only if
OPT ≤ αEv,w [SW(M(v,w ))]. Our goal is to �nd mechanisms that α-approximate the optimal
social welfare for a low α , are DSIC (or BIC), SBB, and ex-post IR (or interim IR).
2.3 Valuation FunctionsWe will consider probability distributions over the following classes of valuation functions. Let
v : 2[k] → R≥0 be a valuation function. �en,
• v is additive if and only if there exist numbers α1, . . . αk ∈ R≥0 such that v (S ) =∑
j ∈S α jfor all S ⊆ [k].
• v is fractionally subadditive (or XOS) if and only if there exists a collection of additive
functions a1, . . . ,ad such that for every bundle S ⊆ [k] it holds that v (S ) = maxi ∈[d] ai (S ).• v is subadditive if and only if for for all S ,T ⊆ [k] it holds that v (S ∪T ) ≤ v (S ) +v (T ).
It is easy to see that every additive function is a XOS function. Further, it is well-known that the
class of submodular functions are contained in the class of XOS functions, and XOS functions are
contained in the class of the subadditive functions.
3 AN INITIAL MECHANISM AND DIRECT TRADE STRONG BUDGET BALANCEBlumrosen and Dobzinski [1] present a mechanism for exchange markets with subadditive valuation
functions. �ey prove the following for this mechanism, which we nameMbd.
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Theorem 3.1 (Blumrosen and Dobzinski [1]). Mechanism Mbd is a DSIC, WBB, ex-post IRrandomized direct revelation mechanism that 4H (s )-approximates the optimal social welfare for com-binatorial exchange markets (n,k,I ,F ) with subadditive valuation functions, where s = min{n, |Ii | :i ∈ [n]} is the minimum of the number of agents and the number of items in an agent’s initialendowment, and H (·) denotes the harmonic numbers.
In particular, this mechanism gives us a constant approximation factor if the number of starting
items of the agents is bounded by a constant.
Now consider a mechanismMsbb that selects an agent i ∈ [n] uniformly at random, runsMbd
on the remaining agents, and allocates the surplus money ofMbd to agent i . We are then able to
prove the following.
Theorem 3.2. MechanismMsbb is DSIC, ex-post IR, SBB, and achieves an 8nH (s )/(n−1)-approximationto the optimal social welfare for exchange markets with subadditive valuations and at least 3 agents.7
Because of space constraints, the proof of this theorem and of various other results in the re-
mainder of this paper, has been omi�ed. �is yields an ex-post IR, SBB, DSIC mechanism that
O (1)-approximates the social welfare if the number of items initially posessed by an agent is
bounded by a constant. �e principle that we used to construct MechanismMsbb can more gener-
ally be used to turn any WBB mechanism into an SBB one, while preserving the DSIC and ex-post
IR properties. It also reveals a problematic aspect of the notion of SBB: it allows for agents to
receive money, while they are not involved in any trade. �is motivates a strengthened notion of
strong budget balance, which we call direct trade strong budget balance.
De�nition 3.3. A mechanism for an exchange market satis�es direct trade strong budget balance(DSBB) if and only if the outcome it generates can be achieved by a set of bilateral trades, where
each trade consists of a reallocation of an item from an agent i to an agent j, and a monetary
transfer from agent j to agent i . Moreover, each item may only be traded once.
DSBB strengthens the traditional SBB notion and seems to be a reasonable requirement in most
two sided markets and exchange markets se�ings. Note that the way in which we strengthen SBB
is rather mild: DSBB still allows an arbitrarily large amount of money to be transfered from one
agent to another as long as at least one item is exchanged in the opposite direction. DSBB does
not even require such a bilateral exchange to be pro�table for both parties, but does nonetheless
seem to rule out the rather unsatisfactory type construction such as the one used inMsbb.
It can be seen that MechanismMsbb does not satisfy DSBB. In the remainder of the paper we
will proceed to design mechanisms for two-sided markets that do satisfy DSBB.8Moreover, two
of our results provide an O (1)-approximation even in se�ings whereMsbb would only provide an
approximation factor of logarithmic order.
4 A MECHANISM FOR UNIT-SUPPLY SELLERS AND XOS BUYERSIn this section we present a DSIC, ex-post IR, and DSBBmechanism for two-sidedmarkets, when sell-
ers initially possess a single item and buyers have XOS valuation functions. �ismechanism achieves
a constant approximation to the optimal social welfare. In this se�ing, we use [k] to denote both the
set of items and the set of sellers, where item j is owned by seller j (so Ij = {j} for all j ∈ [k]). Foreach seller j ∈ [k], we then treat Fj as a distribution overR≥0 instead of a distribution over functions.We assume throughout this section that (n,k,k,I ,G,F ) is a given two-sided market, on which
we run the mechanism to be de�ned. For an allocation (X ,Y ) ∈ A, we shall use the notation
7For two agents, it is straightforward to come up with alternative mechanisms that have the desired properties.
8We note that the double auctions given in [5] also satisfy the DSBB property.
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SWB ,SWSto respectively denote the buyers’ and the sellers’ contribution to the social welfare, i.e.,
SWB (X ,Y ) :=n∑i=1
vi (Xi ),
SWS (X ,Y ) :=k∑i=1
w j (Yj ) =k∑j=1
w j I[j ∈ Yj ].
(1)
Our mechanism requires �xing a price for every item in the market. For a bundle of available
items Λ and an item price vector p = (p1, . . . ,pk ) ∈ Rk≥0, we de�ne the demand correspondence of
buyer i ∈ [n] with valuation function vi as
D (vi ,p,Λ) :=S ⊆ Λ : vi (S ) −
∑j ∈S
pj ≥ vi (T ) −∑j ∈T
pj for all T ⊆ Λ,
i.e.,D (vi ,p,Λ) is the set of bundles of items inΛ that maximise i’s utility under the given item prices.
For a buyer i with valuation function vi , we de�ne the additive representative function for bundle
T ⊆ [k] as any additive function a(vi ,T , ·) : 2[k] → R≥0 such that vi (T ) = a(vi ,T ,T ), and vi (S ) ≥
a(vi ,T ,S ) for all S ⊆ [k]. �e additive representative function of a bundle is guaranteed to exist for
each buyer i and for each valuation function in the support ofGi , by the de�nition of XOS functions.
4.1 MechanismLet A be an algorithm that, given a buyers’ valuation pro�le v and a set of items [k], allocatesthe items to the buyers, without considering the sellers and their valuations. A can either be an
exact algorithm that outputs an optimal allocation of [k] to the buyers (if one is not interested
in the runtime) or an approximately optimal one (in the case that one insists on polynomial-time
implementability). Our mechanism uses A as a black-box for the computation of item prices.
Let X all (v ) = (X all
1(v ), . . . ,X all
n (v )) (where the superscript “all” stands for “allocation”) be the
output allocation of A(v ). Let SW(X all (v )) be the total social welfare of the allocation X all (v ).We de�ne for each item j ∈ [k] its contribution SWB
j (v ) to the social welfare SW(X all (v )) as follows:
if there exists a buyer i that receives item j in allocation X all
i (v ), then SWBj (v ) = a(vi ,X
all
i (v ), {j}).
Otherwise, if j is not allocated to any buyer in X all
i (v ), then SWBj (v ) = 0.
�is notion allows us to make a distinction between high welfare items and low welfare items. Anitem j ∈ [k] is said to have high welfare with respect to SW(X all
i (v )) if and only if Ev [SWBj (v )] ≥
4E[w j ], i.e., the expected social welfare contribution of j if we would allocate j according to X all (v )is at least four times as high as the social welfare that results from leaving item j with its seller.
Formally, let H be the set of high welfare items, i.e., H := {` ∈ [k] : E[SWB` (v )] ≥ 4E[w j ]},
and let L be the set of low welfare items, i.e. L := [k] \ H . For each high welfare item j ∈ H , the
mechanism makes use of the following associated item price pj :
pj :=1
2
Ev [SWBj (v )].
Observe that pj ≥ 2E[w j ] for all j ∈ H , by our de�nition of high welfare items.
�e reason why H is chosen in such a way is twofold: �rst, the items in L if kept by their sellers
provide a welfare loss of at most a constant factor; second, every item in H is guaranteed to be sold
(if sold) at a high price, to make sure that the buyer receiving the item has a high valuation for it.
Our (randomized) mechanism does the following. First, it considers the sellers with an item in
H (in any order) and asks each of them whether they would sell their item for a price of pj . As
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Approximately E�icient Two-Sided Combinatorial Auctions 1
mentioned above, by de�nition of the prices, every seller j ∈ H accepts the price with probability
at least 1/2, by Markov’s inequality (recall that pj ≥ 2E[w j ] for all j ∈ H ).
Tomake sure that this probability is exactly 1/2, the seller j is only given the opportunity to sell heritem at the pricepj with probabilityqj such that (in expectation) the o�er is acceptedwith probabilityexactly 1/2. Formally this means that the mechanism makes an o�er to the seller j with probability
qj :=1
2Fj (pj ), where Fj (pj ) = Pr[w j ≤ pj ].
An item inH is considered to be “in the market” if the corresponding seller accepts the mechanism’s
o�er. A�er the mechanism has made the o�ers to the sellers of H , it knows which items are in the
market and then asks each buyer (sequentially, in any order) for her favorite bundle of items among
those items that are still in the market. If an item j gets requested by a buyer, then j is transferredfrom its corresponding seller j, and the buyer pays pj to seller j. Item j is then removed from the
set of items in the market, and the mechanism proceeds to the next buyer.
We call the mechanism sketched aboveM1-supply, which we now present more precisely:
(1) Let H := {j ∈ [k] : Ev [SWBj (v )] ≥ 4E[w j ]}.
(2) For all j ∈ H , set pj :=1
2Ev [SWB
j (v )].
(3) Let Λ1 := ∅,Xi := ∅ for all i ∈ [n] and Yj := {j} for all j ∈ [k].(4) For all j ∈ H :
(a) Set qj := 1/(2Pr[w j ≤ pj ]).(b) With probability qj , o�er payment pj in exchange for her item.
(c) If j accepts the o�er, set Λ1 := Λ1 ∪ {j}.(5) For all i ∈ [n]:
(a) Buyer i chooses a bundle Bi ∈ D (vi ,p,Λi ) that maximises her utility.
(b) Allocate the accepted items to buyer i , i.e., Xi := Bi and Yj := ∅ for all j ∈ Bi .(c) Remove the selected items from the available items, i.e., Λi+1 := Λi \ Bi .
(6) Return the outcome consisting of allocation (X = (X1, . . . ,Xn ),Y = (Y1, . . . ,Yk )) and payments
ρ = (ρB ,ρS ), where ρBi =∑j ∈Xi pj for i ∈ [n] and ρ
Sj = −pj I[Yj = ∅] for j ∈ [k].
Note that Algorithm A is only used in the �rst steps of mechanismM1-supply, where Ev [SWBj (v )]
is computed. Let α be the factor by which A is guaranteed to approximate the social welfare of
the buyers.
4.2 ResultsNow, we are ready to present the main result of this section:
Theorem 4.1. M1-supply is ex-post IR, DSIC, DSBB, and (2 + 4α )-approximates the optimal socialwelfare.
In particular, taking for A an optimal algorithm (i.e., α = 1), we obtain that there exists a
mechanism that is ex-post IR, DSIC, DSBB, and 6-approximates the optimal social welfare. As
mentioned in Section 1.3, one may alternatively take for A a polynomial time α-approximation
algorithm and use the technique of [11], to obtain a mechanism with runtime POLY (1/ϵ ,n,m) thatapproximates the optimal social welfare within a 2+ 4α multiplicative factor and an ϵ additive term.
We split the proof of �eorem 4.1 into two lemmas that separately bound the sellers’ and the buy-
ers’ relative contributions to the social welfare. We use the notation OPT as de�ned in Section 2, and
we use ALG to denote the expected social welfare of the mechanism, i.e., Ev,w [SW(M1-supply (v,w ))].Moreover, the superscripts S ,B respectively denote the sellers’ and buyers’ contributions to the social
welfare, e.g., OPT = OPTS + OPTB and ALG = ALG
S + ALGB, consistent with the notation of (1).
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�e following lemma is a simple consequence of the fact that M1-supply lets every seller in Haccept an o�er with probability exactly 1/2.
Lemma 4.2. If every seller j ∈ H puts her item into the market with probability exactly 1/2, then
2ALGS ≥
k∑j=1
E[w j ] ≥ OPTS .
Proof. �e second inequality is trivial, so we focus on the �rst inequality. First, observe that
Pr[w j > pj ] ≤ Pr[w j > 2E[w j ]] <1
2
,
where the �rst inequality holds because j ∈ L, and the second inequality follows by Markov’s
inequality. �us, with probability at least 1/2 a seller j is happy to sell her item at price pj . But everyseller receives an o�er from the mechanism with probability qj := 1/(2Pr[w j ≤ pj ]), so every seller
in H accepts to trade with probability exactly 1/2. �is implies that every seller j ∈ H contributes
in expectation at least E[w j ]/2 to the social welfare. Moreover, every seller in L never trades, so
that such a seller contributes her full expected valuation to the expected social welfare. �
Next, we provide a more di�cult bound that relates ALGBand ALG
Sto OPT
B.
Lemma 4.3. �e buyers’ contributions to the optimal social welfare is bounded by
4αALGB + 4αALGS ≥ OPTB .
Intuitively, Lemma 4.3 uses two main ingredients:
• the partition of the items between high-welfare (H ) items and low-welfare items (L), and• the de�nition of SWB
j (v ) w.r.t. a one-sided (approximation) algorithm A.
�e la�er tells us that the sum of the expected contributions of all the items, i.e.
∑kj=1 E[SW
B` (v )],
is an upper-bound on OPTB/α . From the former we know that:
• the sellers do not trade items in L, and this is enough to ensure that their contribution to
the expected social welfare is greater than a constant fraction of the expected contribution
of the items in L, i.e. ALGS > 1
4
∑j ∈L E[SWB
j (v )]. Moreover,
• the only items that the agents can trade are those that have a high welfare w.r.t. SW(X all
i (v )).From that we can infer that the contribution of the buyers to the expected social welfare
is greater than a constant fraction of the expected contribution of the items in H , i.e.
ALGB > 1
4
∑j ∈H E[SWB
j (v )].
By combining these bounds, the claim of Lemma 4.3 follows. �eorem 4.1 is then obtained
straightforwardly from Lemma 4.2 and Lemma 4.3.
Proof of Theorem 4.1. �e bound on the approximation ratio follows from the sum of the
inequalities of Lemma 4.2 and Lemma 4.3. Moreover, it is a dominant strategy for a seller to accept
if and only if the payment o�ered to her exceeds her valuation, and it is a dominant strategy for
a buyer to choose a utility-maximising bundle for the items and item prices o�ered to her. �us,
when viewed as a direct revelation mechanism,M1-supply is DSIC. It is clear that participating in
the mechanism can never lead to a decrease in utility for both buyers and sellers, and therefore
the mechanism is also ex-post IR. Lastly, it is straightforward to see that the mechanism is DSBB,
as the de�nition ofM1-supply which we gave in terms of sequential posted pricing naturally yields
us the required set of bilateral trades. �
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Approximately E�icient Two-Sided Combinatorial Auctions 1
5 A MECHANISM FOR ADDITIVE SELLERS AND XOS BUYERSWe now consider the se�ing in which sellers may own multiple distinct items and have an additive
valuation function over them. We design a DSBB mechanism that is DSIC and ex-post IR on the
sellers’ side, and BIC and interim IR on the buyers’ side. At the end of the section we show that, in
the case that both buyers and sellers have additive valuation functions, the mechanism we present
is DSIC and ex-post IR on both sides of the market.
We assume throughout this section that (n,m,k,I ,G,F ) is a given two-sidedmarket with XOS buy-
ers and additive sellers, on which we run the mechanism to be de�ned. Like in the previous section,
the buyers are still assumed to have XOS valuation functions over the items. Since now the number
of items and sellers is di�erent in general, we usem to denote the number of sellers and k for the
number of items. �e valuationw j of a seller j is now an additive function. We reuse the following
notation from Section 4: the allocation (X all
1(v ), . . . ,X all
n (v )) returned by an allocation algorithm Aon inputv returns an allocation of [k] to [n]. We let α ≥ 1 again denote the approximation factor by
whichA approximates the social welfare. For XOS valuationvi and bundleT ⊆ [k] we use a(vi ,T , ·)to denote the additive representative function of vi for T . Also we use the buyers’ social welfare
contribution SWB` (v ) for item ` ∈ [k] and buyers’ valuation pro�lev , as de�ned in Section 4.
Furthermore, we de�ne the sellers’ social welfare contribution SWS`(w ) for item ` ∈ Ij and sellers’
valuation pro�le w as SWS`(w ) := w j ({`}). Due to the fact that for j ∈ [m], w j is an additive
function, there is no need for de�ning the notion of an additive representative function for a seller.
5.1 MechanismWe aim to design a BIC, interim IR, and DSBB mechanism that approximates the optimal social
welfare within a constant. We propose the following mechanism, which we refer to asMadd. We let
Hj := {` ∈ Ij : E[SWB` (v )] ≥ 4E[SWS
`(w )]} and Lj := Ij \Hj for all j ∈ [m], and we letH :=
⋃mj=1Hj
and L := [k] \ H denote the sets of high-welfare items and low-welfare items, respectively. Our
mechanism will only allow trading items in H . We de�ne for ` ∈ H the item price
p` :=1
2
E[SWB` (v )],
similar to what we did forM1-supply.
An essential di�erence betweenMadd andM1-supply is that the order in which buyers and sellers
are processed is reversed. Mechanism Madd roughly works as follows. It �rst asks every buyer
which set of items it would like to receive from those items in H that have not been requested
yet. �en Madd o�ers every seller j ∈ [m] a payment in exchange for the subset of all items in
Ij that have been requested. �is o�er is made with probability qj , chosen in such a way that
the requested items of seller j are transferred to the buyers with probability 1/2. �e items of
the sellers accepting the o�er are transferred to the buyers for the corresponding item prices.
Buyers act strategically, and will request a bundle of items that maximises their expected utility,
knowing that the item sets requested from each seller will be assigned to them with probabil-
ity 1/2.9 In our mechanism, the sellers will each have a dominant strategy, while the buyers’
aformentioned behaviour relies on the sellers playing their dominant strategies. �is reliance
results in a BIC (rather than a DSIC) mechanism. Below we describe the mechanism in more
detail and we subsequently provide an example of the mechanism’s execution on a simple in-
stance.
9Buyers may need to make complex calculations in order to establish which bundle maximises her expected utility.
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(1) For ` ∈ [k], compute E[SWB`(v )] and E[SWS
`(w )].
(2) For all j ∈ [m], compute Hj .
(3) Compute H and L.(4) Let Λ1 := H , Xi := ∅ for all i ∈ [n], and Yj := Ij for all j ∈ [m].
(5) For each buyer i ∈ [n]:(a) Ask buyer i to select an expected-utility maximising bundle Bi ⊆ Λi given the prices
{p` : ` ∈ Λi } from the set of available items Λi (where the expectation is taken w.r.t. the
randomness of the valuations and the mechanism).
(b) Update the set of available items Λi+1 := Λi \ Bi .(6) Let B :=
⋃ni=1 Bi be the set of all items demanded by the buyers.
(7) For each seller j ∈ [m]:
(a) Let Sj := B ∩ Hj be the set of items owned by seller j that are demanded.
(b) Let p (Sj ) :=∑
`∈Sj p` and let qj = 1/(2Pr[w j (Sj ) ≤ p (Sj )]).
(c) With probability qj , o�er payment p (Sj ) in exchange for the bundle Sj . Otherwise, skip this
seller.
(d) If the seller accepts the o�er, allocate each items in Sj to the buyer that requested it (i.e.,
remove Sj from Yj and add Sj ∩ Bi to Xi for all i ∈ [n])(8) Return the outcome consisting of allocation (X = (X1, . . . ,Xn ),Y = (Y1, . . . ,Yk )) and payments
ρ = (ρB ,ρS ), where ρBi =∑
`∈Xi p` for i ∈ [n] and ρSj =∑
`∈Ij \Yj −p` for j ∈ [m].
Notice the mechanismMadd runs in polynomial time, but it makes use of a variant of a standard
demand query in which the mechanism gives prices for the items, and asks a buyer which bundle
she would like if, for each item in that bundle, she were to receive it with probability 1/2. �is places
a heavier computational and cognitive burden on the agent than with standard demand queries.
We will not address in the present paper the complexity aspects of the buyer’s task to answer such
queries, though we believe that it is an interesting open question to investigate.
�e following example illustrates some important aspects of Madd, and the strategies of the
buyers under a BNE.
Example 5.1. �ere is one buyer and two unit-supply sellers. Each seller has one item. �e
buyer has two XOS valuation functions v1 and v2, each chosen with probability 1/2. Valuation v1is composed of 3 additive functions a1, a2, and a3, i.e., v1 (S ) = max{a1 (S ),a2 (S ),a3 (S )}. Valuationv2 consists of a single additive function a4. Each seller j has a valuation function w j = 0. Recall
that a function a is additive if there exists α1, . . . ,αk such that a(S ) =∑
j ∈S α j for all S ⊆ [k]. �e
functions a1 to a4 are described in the table below by listing the values α1 and α2.
Function item 1 (α1) item 2 (α2)
a1 0 2
a2 8 0
a3 7 2
a4 1 6
Let us compute the prices o�ered by the mechanismMadd when A is an optimal algorithm. �us,
we need to compute the expected contribution to the optimal social welfare of every item. First,
notice that the optimum allocates the items 1 and 2 to the buyer when her valuation is v1. In this
case the contribution to the optimal social welfare of item 1 is 7, and the contribution of item 2
is 2. Similarly, if the buyer has valuation v2, the optimum still allocates items 1 and 2 to her, but
in this case the contribution to the optimal social welfare of item 1 is 1, and the contribution of
item 2 is 6. �us, the expected contribution of every item to the optimal social welfare is 4, i.e.,
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Approximately E�icient Two-Sided Combinatorial Auctions 1
E[SWBj (v )] = 4 for all j = 1,2. Since the price pj of each item is de�ned to be half of the expected
contribution to the optimal social welfare, pj = 2 for all the items.
When the mechanism asks a buyer to select a bundle that maximizes her expected utility, the
buyer has to answer by taking into account the fact that each item in her requested bundle will
be allocated with probability 1/2. First, consider the case when the buyer has valuation v1. In this
case the expected utility for the di�erent bundles are:
u ({1}) =1
2
· (8 − 2) +1
2
· 0 = 3,
u ({2}) =1
2
· (2 − 2) +1
2
· 0 = 0,
u ({1,2}) =1
4
· (8 − 2) +1
4
· (2 − 2) +1
4
· (9 − 4) +1
4
· 0 =11
4
.
�e utility-maximising bundle that will be requested by the buyer in case of v1 is {1}, in which case
the mechanism will let the buyer pay Seller 1 a price of 2 in exchange for the item. Instead, if the
valuation of the buyer is v2, then the requested bundle will be {1,2}, in which case the mechanism
will let the buyer pay both sellers a price of 2 in exchange for their items.
5.2 ResultsOur main result forMadd is the following theorem.
Theorem 5.2. �emechanismMadd is interim IR, BIC, DSBB, and (2+4α )-approximates the optimalsocial welfare.
By taking for A an optimal algorithm (i.e., α = 1), we obtain a mechanism that is ex-post IR,
DSIC, SBB, and 6-approximates the optimal social welfare. Again, we split the proof of this theorem
5.2 into two lemmas that separately bound the sellers’ and the buyers’ relative contributions to
the social welfare. Like the previous section, we use the notation OPT as de�ned in Section 2, and
we use ALG to denote the expected social welfare of the mechanism. Moreover, we use again the
superscripts B and S to refer to the buyers’ and sellers’ expected contribution to the social welfare
of a given allocation, as we did in Section 4.
Let us �rst discuss how we bound the sellers’ expected contribution to the optimal allocation.
Lemma 5.3.
2ALGS ≥ OPT
S .
Proof. �e only items that our mechanisms potentially reallocates are the ones belonging to
H . Every item in L stays with its seller. For the items in H , the mechanism ensures every seller
sells her demanded bundle with probability exactly 1/2, so for each seller it holds that she retains
her full initial endowment with probability at least 1/2, which implies the claim. �
Similarly, we want to provide an upper bound on the buyers’ expected contribution to the optimal
allocation. To do that we need two auxiliary propositions.
�e �rst proposition exploits the partition of the items among high-welfare items and low-welfare
items. Since the low-welfare items are not traded, the sum of the expected contribution of the buyers
on the high-welfare items and the expected contribution of the sellers on the low-welfare items
gives us an upper bound on the buyers’ expected contribution in the allocation computed by A.
Proposition 5.4.∑`∈H
Ev [SWB` (v )] + 4
∑`∈L
Ew [SWS` (w )] >
n∑i=1
Ev [vi (Xall
i (v ))].
ACM Transactions on Economics and Computation, Vol. 1, No. 1, Article 1. Publication date: January 2017.
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1 R. Colini-Baldeschi, P.W. Goldberg, B. de Keijzer, S. Leonardi, T. Roughgarden, and S. Turche�a
Proof. Let a(vi ,Xall
i (v ), ·) be the representative additive function of vi for the bundle Xall
i (v ).�en,
n∑i=1
E[vi (Xall
i (v ))] =
n∑i=1
E
∑`∈X all
i (v )
a(vi ,Xall
i (v ), {`})
=
n∑i=1
k∑`=1
E[a(vi ,Xall
i (v ), {`})I[` ∈ X all
i (v )]]
=
k∑`=1
E[SWB` (v )]
=∑`∈H
E[SWB` (v )] +
∑`∈L
E[SWB` (v )]
<∑`∈H
E[SWB` (v )] + 4
∑`∈L
E[SWS` (w )].
�e last inequality follows because by de�nition of L,
4
∑`∈L
E[SWS` (w )] >
∑`∈L
E[SWB` (v )].
�
Now, since buyers can obtain only high-welfare items, their contribution to the expected social
welfare ofMadd is greater than a constant fraction of the expected contribution of the high-welfare
items to the allocation computed by A.
Proposition 5.5.
ALGB ≥
1
4
∑`∈H
Ev [SWB` (v )].
�us, using Proposition 5.4 and Proposition 5.5 we can prove that the sum of the buyers’ expected
contribution and the sellers’ expected contribution ofMadd provides a constant approximation to
the buyers’ expected contribution in the optimal allocation.
Lemma 5.6.
4αALGB + 4αALGS > OPTB .
Proof. By Proposition 5.5, 4ALGB ≥
∑`∈H Ev [SWB
` (v )]. Moreover, our mechanism leaves
every item ` ∈ L with its seller, and so ALGS ≥∑
`∈L Ew [SWS`(w )]. �erefore,
4ALGB + 4ALGS ≥
∑`∈H
Ev [SWB` (v )] + 4
∑`∈L
Ew [SWS` (w )] >
n∑i=1
Ev [vi (Xall
i (v ))] ≥1
αOPT
B .
�e second inequality holds by Proposition 5.4, and the last inequality follows because we de�ned αto be the approximation factor of algorithm A, which is the algorithm that we assumed to generate
allocation X all (v ). �
Finally, we are ready to prove the main theorem of this section.
Proof of Theorem 5.2. On the sellers’ side, the mechanism is ex-post IR and DSIC: the sellers
solely have to decide between accepting or rejecting a single o�er to receive a proposed payment
in exchange for a bundle of items, and it is clearly a dominant strategy to accept if and only if
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Session 9a: Auctions -- Equilibrium EC'17, June 26–30, 2017, Cambridge, MA, USA
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Approximately E�icient Two-Sided Combinatorial Auctions 1
such an exchange leads to an improvement in the seller’s utility. Every buyer chooses a bundle
that maximises her expected utility, and this choice depends solely on the choice of strategies of
the sellers. �erefore, the mechanism has a BNE in which the sellers play a dominant strategy,
and the mechanism is thus ex-interim IR and BIC. �e fact that the mechanism is DSBB follows
from its de�nition, which makes clear that payments are de�ned by the appropriate sequence of
trades and payments from buyers to sellers. �e approximation guarantee follows by the sum of
the inequalities of the above Lemmas 5.3 and 5.6. �
It is important to notice that the mechanismMadd turns into a DSIC and ex-post IR mechanism
if the buyers have additive valuations instead of XOS valuations.
Corollary 5.7. For the special case that for all i ∈ [n], distributionGi is over additive valuationfunctions,Madd is ex-post IR, DSIC, DSBB and (2 + 4α )-approximates the optimal social welfare.
Proof. If a buyer i ∈ [n] has an additive valuation function, it is a dominant strategy to request
the items in Λi (v<i )) for which it holds that vi ({`}) > p` . �is follows from the simple fact that by
additivity, the utility that a buyer has for any bundle of items S can be wri�en as
∑`∈S vi ({`}) − p` .
�us, for every item ` ∈ [k] that a buyer requests (recall that this item is then allocated to her for
price p` with probability 1/2), a term of (1/2) (vi ({`}) − p` ) gets added to her expected utility. So
including ` in her requested bundle is pro�table if and only if vi ({`}) − p` ≥ 0. Using the same
argument, the ex-post IR property is also satis�ed by following this strategy. �
6 DISCUSSIONAn open problem is to extend or re�ne our mechanisms so that they satisfy the DSIC and ex-post
IR properties for the case of XOS buyers and additive sellers. �e �rst naive approach for doing so
might be trying to consider every additive seller as a set of distinct unit-supply sellers and then run
M1-supply. However, this is not guaranteed to work due to the fact that an additive valuation function
may have intrinsic interdependencies among the items (e.g. if there are duplicates among the items)
and so the independence of these distinct unit-supply sellers is not guaranteed. Something we might
additionally consider to do is to ask every seller for her favorite bundle to place in themarket, yet this
may cause a seller to regret having chosen that particular bundle a�er seeing the realizations of the
buyers’ valuations. On the other hand, it also seems highly challenging to establish any sort of im-
possibility result for any reasonably de�ned class of posted price mechanisms for two-sided markets.
Another natural direction is to extend the above mechanism to the se�ing in which both buyers
and sellers possess an XOS valuation function over bundles of items. A �rst challenge consists
in �nding a suitable de�nition of the sellers’ social welfare contribution of an item using the
corresponding representative additive function.
ACKNOWLEDGMENTS�e authors thank anonymous referees for their helpful comments. Tim Roughgarden is supported
by the NSF Grant CCF-1524062. Stefano Leonardi is partly supported by the Google Focused Award
on “Algorithms for Large-scale Data Analysis”. Part of this work was done while Stefano Leonardi
and Tim Roughgarden were visiting the Economics and Computation semester at the Simons
Institute for the �eory of Computing.
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1 R. Colini-Baldeschi, P.W. Goldberg, B. de Keijzer, S. Leonardi, T. Roughgarden, and S. Turche�a
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